# region

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## 1—10 of 60 matching pages

##### 1: 28.17 Stability as $x\to \pm \mathrm{\infty}$

###### §28.17 Stability as $x\to \pm \mathrm{\infty}$

… ►For real $a$ and $q$ $(\ne 0)$ the stable regions are the open regions indicated in color in Figure 28.17.1. The boundary of each region comprises the*characteristic curves*$a={a}_{n}\left(q\right)$ and $a={b}_{n}\left(q\right)$; compare Figure 28.2.1. … ►

##### 2: 12.20 Approximations

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►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $U(a,b,x)$ and $M(a,b,x)$ (§13.2(i)) whose regions of validity include intervals with endpoints $x=\mathrm{\infty}$ and $x=0$, respectively.
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##### 3: 10.72 Mathematical Applications

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►In regions in which (10.72.1) has a simple turning point ${z}_{0}$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and ${z}_{0}$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)).
These expansions are uniform with respect to $z$, including the turning point ${z}_{0}$ and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities.
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►In regions in which the function $f(z)$ has a simple pole at $z={z}_{0}$ and ${(z-{z}_{0})}^{2}g(z)$ is analytic at $z={z}_{0}$ (the case $\lambda =-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm \sqrt{1+4\rho}$, where $\rho $ is the limiting value of ${(z-{z}_{0})}^{2}g(z)$ as $z\to {z}_{0}$.
These asymptotic expansions are uniform with respect to $z$, including cut neighborhoods of ${z}_{0}$, and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation.
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##### 4: 21.10 Methods of Computation

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##### 5: Bonita V. Saunders

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►This work has resulted in several published papers presented as contributed or invited talks at universities and regional, national, and international conferences.
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##### 6: 5.10 Continued Fractions

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5.10.1
$$\mathrm{Ln}\mathrm{\Gamma}\left(z\right)+z-\left(z-\frac{1}{2}\right)\mathrm{ln}z-\frac{1}{2}\mathrm{ln}\left(2\pi \right)=\frac{{a}_{0}}{z+}\frac{{a}_{1}}{z+}\frac{{a}_{2}}{z+}\frac{{a}_{3}}{z+}\frac{{a}_{4}}{z+}\frac{{a}_{5}}{z+}\mathrm{\cdots},$$

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##### 7: 14.31 Other Applications

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►The conical functions ${\U0001d5af}_{-\frac{1}{2}+\mathrm{i}\tau}^{m}\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)).
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##### 8: 33.23 Methods of Computation

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►Bardin et al. (1972) describes ten different methods for the calculation of ${F}_{\mathrm{\ell}}$ and ${G}_{\mathrm{\ell}}$, valid in different regions of the ($\eta ,\rho $)-plane.
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►Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for ${F}_{0}$ and ${G}_{0}$ in the region inside the turning point: $$.

##### 9: 19.7 Connection Formulas

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►The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of $\mathrm{\Pi}(\varphi ,{\alpha}^{2},k)$ when ${\alpha}^{2}>{\mathrm{csc}}^{2}\varphi $ (see (19.6.5) for the complete case).
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►The second relation maps each hyperbolic region onto itself and each circular region onto the other:
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►The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other:
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##### 10: 19.21 Connection Formulas

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►Change-of-parameter relations can be used to shift the parameter $p$ of ${R}_{J}$ from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)).
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►For each value of $p$, permutation of $x,y,z$ produces three values of $q$, one of which lies in the same region as $p$ and two lie in the other region of the same type.
In (19.21.12), if $x$ is the largest (smallest) of $x,y$, and $z$, then $p$ and $q$ lie in the same region if it is circular (hyperbolic); otherwise $p$ and $q$ lie in different regions, both circular or both hyperbolic.
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